The video problem occurs on both my computers, one is running Windows 7 and the other is Ubuntu 13.04. Both are running the latest Firefox version. I haven't installed any addons for either computer. I updated Flash Player also. Seems to be a common problem out there, sounds like a bug in Firefox maybe.
Edit:
The problem was with Adobe Flashplayer. I downloaded the Flashplayer uninstaller and followed the instructions for troubleshooting Firefox problems and cleaned up the Flashplayer leftovers. Rebooted. Reinstalled Flashplayer and now YouTube videos work just fine in Firefox and Internet Explorer on my Win7 machine. I'll tackle the Linux machine later on. Thanks uchihaMadara!